I like the explanation, but there's no reason to skip over the complex exponents and Euler's formula like that. They're not really that hard to understand intuitively: think of all multiplication as a continuous process. Then f = e^x is simply the function that transforms the multiplicative identity (1) into ln(x).
Substitute "-1" into the left side of that equation, and see that no real value of x will suffice. This is related to the fact that the imaginary constant (i) wasn't discovered, it was simply declared as an unknown quantity that squares to -1.
The real magical part is that i still works in more complicated situations: multiplying any real number by e^(ix) as x increases gradually transforms it into an imaginary number, and then into its own negative, behaving like a counter-clockwise rotation when visualized in the complex plane.
Substitute "-1" into the left side of that equation, and see that no real value of x will suffice. This is related to the fact that the imaginary constant (i) wasn't discovered, it was simply declared as an unknown quantity that squares to -1.
The real magical part is that i still works in more complicated situations: multiplying any real number by e^(ix) as x increases gradually transforms it into an imaginary number, and then into its own negative, behaving like a counter-clockwise rotation when visualized in the complex plane.